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The answer, it turns out (thanks as always to our readers who responded in great numbers and mostly without unnecessary snark), is actually pretty simple: That’s likely a flashback of some kind to Black Widow’s childhood.
Part of the elaborate brainwashing that Natasha was subjected to over the years that made her into Black Widow actually led her to believe that she’d been raised as a Russian ballerina — an explanation of sorts for some of the skills she would use in more deatly ways later, and a rationale for training her in some of those skills.
To which she responded, “You know I have.”
That line implied that much of her traumatic and complex backstory had been carried over from the comics, but it did so without parsing out specifically which elements of that backstory those were. It seems that Joss Whedon — ever the champion of women kicking ass — wants to explore a bit of that in the second film, and that’s a surprise to no one, really.
Could these things all be connected? If the characters were somehow reliving parts of their past in the film, it would explain Captain America’s flashback and Black Widow’s, as well as Thor being hammer-less (as he was in much of Thor) and Hulk being out of control (as he was in every screen appearance excluding The Avengers).